Question

Let $y=mx+c$ is a tangent at point $P$ to the curve $y^2=2x,$ meets the curve again at $Q.$ Then which of the following condition holds true

• A. $mc<1$
• B. $2mc<1$
• C. $mc>1$
• D. $2mc>-1$

Answer 1

The correct option is B $2mc<1$

As, $y=mx+c$ is tangent at $P$ and intersect the curve at $Q.$
Now, solving the equation of tangent with the equation of curve.
$\Rightarrow (mx+c{)}^2=2x$
$\Rightarrow m^2{x}^2+2x(mc-1)+c^2=0$
Above equation must have $2$ real and distinct roots.
$\therefore D>0$
$\Rightarrow 4(mc-1{)}^2-4m^2{c}^2>0$
$\Rightarrow 4m^2{c}^2-8mc+4-4m^2{c}^2>0$
$\Rightarrow 2mc<1$